INITIAL APPROXIMATION FOR NEWTON-RAPHSON ITERATION … · INITIAL APPROXIMATION FOR NEWTON-RAPHSON...
-
Upload
nguyenkhanh -
Category
Documents
-
view
259 -
download
4
Transcript of INITIAL APPROXIMATION FOR NEWTON-RAPHSON ITERATION … · INITIAL APPROXIMATION FOR NEWTON-RAPHSON...
INITIAL APPROXIMATION FOR
NEWTON-RAPHSON ITERATION TO CALCULATE
THE NEUTRAL AXIS POSITION OF
GLULAM BAMBOO
Inggar Septhia Irawati1, Bambang Suhendro
2, Ashar Saputra
3
,Tibertius Agus Prayitno4
1 Doctoral Student, Civil and Environmental Engineering Department, Gadjah Mada University, Jl. Grafika
No 2 Yogyakarta, Indonesia, Tel.+62 274545675, Fax. +62 274545676,
Email:[email protected] 2 Lecturer, Civil and Environmental Engineering Department, Gadjah Mada University, Jl. Grafika No 2
Yogyakarta, Indonesia, Tel.+62 274545675, Fax. +62 274545676, Email: [email protected] 3 Lecturer, Civil and Environmental Engineering Department, Gadjah Mada University, Jl. Grafika No 2
Yogyakarta, Indonesia, Tel.+62 274545675, Fax. +62 274545676, Email: [email protected] 4 Lecturer, Wood Science and Technology Department, Gadjah Mada University, Jl. Agro No 1 Yogyakarta,
Indonesia, Tel.+62 2746491428, Fax. .+62 274550541, Email: [email protected]
Received Date: July 9, 2012
Abstract
Glued laminated (glulam) bamboo has possibility to be used as construction material. For structural
purpose, an analytical calculation is needed to estimate the structure capacity. As bamboo typically
has non linear material behavior under compression parallel to grain, it is essential to figure out the
non linear behavior of glulam bamboo for structural component.
The research purpose is to figure out non linear behavior of glulam bamboo by determining
maximum capacity under axial and lateral loadings according to actual properties of tension and
compression parallel to grain. The failure load is numerically computed based on finite different
method. A Newton-Raphson iterative procedure is employed to determine the deflection curve and
neutral axis position that meet the equilibrium condition. Assumption used is that tensile or
compressive stress is equal to zero when it is greater than its ultimate stress. This paper is focused
to describe iterative procedure to obtain neutral axis position. The novelty of this study is the
proposed initial approximation for iteration of neutral axis position which has never strongly been
considered by previous study. Its application and detail discussion are presented. There are two
reasons why initial approximation for obtaining neutral axis position is important to be understood.
First, it influences the speed of convergence, and secondly, when the internal stress is close to the
maximum strength, it might influence the convergence of calculation.
Keywords: Glulam Bamboo, Initial Approximation, Neutral Axis Position, Newton-Raphson
Iteration, Non Linear Behavior
Introduction
Glued laminated (glulam) bamboo is made of several bamboo strips bonded together with
adhesive (Figure 1). Based on bamboo strips layout, there are two type of glulam bamboo,
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.41
Figure 1. Cross Section: (a) vertically glulam bamboo; (b) horizontally glulam bamboo
Figure 2. General numerical procedure
i.e., vertically and horizontally glulam bamboo. Glulam bamboo is comparable to other
wood product [1]. It has possibility to be used as structural component, i.e., column as well
as structural beam [1]. Glulam bamboo also has similar problem to wood product [1]. It
needs to be treated for fire and insects [1]. However, it is more eco-friendly material
compared to other glulam timber. Bamboo as a raw material is more renewable material. It
can grow quickly. It reaches structural grade material within 3 – 5 years whereas timber
achieves structural grade material within 20 years [2]. Further, when glulam bamboo is
used for structural purpose, an analytical calculation is needed to estimate the structure
capacity.
Bamboo has non linear material behavior under compression parallel to grain. In spite
of this, the structural capacity is usually calculated by using assumption that bamboo is a
linear material. Accordingly, it is essential to figure out the non linear material behavior of
glulam bamboo for structural component. For this reason, the research purpose is to
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.42
understand non linear behavior of glulam bamboo beam-column by determining maximum
capacity based on actual properties of tension and compression parallel to grain. Numerical
program for calculating failure load of glulam bamboo subjected to axial and lateral load
has being developed based on finite difference method. Structural member is divided into
several nodal points that have equal span. Bending failure load is obtained by calculating
the equilibrium deflected shape of the structure. Shear failure load is obtained by
comparing the internal shear stress towards shear strength. In general, the developed
numerical program is illustrated in Figure 2.
Figure 3. Procedure for obtaining equilibrium deflected shape
One of the important steps for calculating an equilibrium deflected shape is the
determination of neutral axis position. The neutral axis position is defined by obtaining an
equilibrium condition for axial forces at each nodal point. Concerning glulam bamboo
axially and laterally loaded, the neutral axis position will vary along the beam depending
on deflection, internal axial force as well as internal moment at each cross section. An
iterative procedure is needed to determine neutral axis position. Accordingly, Newton-
Raphson method is employed for this iteration. It is useful technique for finding zeros of
function [3]. However, a question arises with respect to choosing initial approximation.
Previous procedure for calculating neutral axis position has been published [4; 5; 6] but
initial approximation has not been clearly stated. It is important to be known for the reason
that it will greatly influence the speed of convergence [7]. Besides that, by using improper
initial approximation, non convergent result may be occurred during calculation [8]. In
case of glulam beam subjected to axial and lateral load, non convergent result may be
occurred when the internal stress is close to the maximum strength. Thus, this paper
presents observation process as well as its result to identify the proper initial value to
calculate neutral axis position of glulam bamboo by using Newton-Raphson method.
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.43
Equilibrium Deflected Shape
Iterative procedure to obtain equilibrium deflected shape is illustrated in Figure 3, where
MAXITDFL is the maximum number of iteration for obtaining equilibrium deflected shape;
U and V are deflection in X and Y direction, respectively; CU and CV are the correction of
deflection in X and Y direction, respectively; M is number of nodal point. Material
properties that consist of tension and compression parallel to grain, beam dimension,
loading configuration, number of nodal point, number of Gauss point and coefficients
needed are defined. A number of assumptions are employed in the calculation, as follow:
Plane sections before bending remain plane upon flexure.
Figure 4. Coordinates system: (a) cartesian coordinate system; (b) natural coordinate
system
The study is focused on the non linear material of glulam bamboo. The relation
between curvature Φ and deflection V that is written in Equation (1) is simplified
into Equation (2) [9]. It can be rewritten that curvature is presented by the second
derivatives of deflection.
2/32 )'1(
"
V
V
(1)
"V (2)
Tensile as well as compressive stresses parallel to grain are equal to zero when they
are greater than the ultimate stress. Glulam bamboo is assumed as homogeneous
material.
Lateral loads act through the centroid of the cross section so that the twist moment
is not considered.
Beam is supported by simple bearings.
The initial assumption of horizontal deflection U and vertical deflection V for nodal i=2
to i=M-1 are obtained by using Equation (3) and Equation (4).
0iU (3)
1416.3sin
2000 L
zLV im
i (4)
where zmi is the distance of nodal point i measured from point 0, and L is the beam length.
Coordinate system can be seen in Figure 4(a). External moments for nodal point i=2 to
i=M-1 are calculated by using the Equation (5) and Equation (6).
NLk
kkpimkyiLx
im
xim
xix zzPVAXFML
zM
L
zMM
100
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.44
NPj
jjpjy
im zLPL
z
1
(5)
NLk
kkpimkxiLy
im
yim
yiy zzPUAXFML
zM
L
zMM
100
NPj
jjpjx
im zLPL
z
1
(6)
where Mxi, Mx0 and MxL are external moments in X direction at nodal point i, 0 and L,
respectively; Myi, My0 and MyL are external moments in Y direction at nodal point i, 0 and L,
respectively; zpj is the distance of lateral load j that is located at the left side from nodal
Figure 5. Figure denotation
point i and measured from point 0; Pyj and Pxj are lateral loads in Y and X direction,
respectively; NP is the number of lateral load; NL is the number of lateral load located on
the left side from nodal point i; AXF is external axial force. Zero point is located on the left
bearing, as shown in Figure 5.
Internal moments of cross sections located at nodal point i=2 to i=M-1 are calculated
by using Equation (7) and Equation (8).
dd,
1
1
1
1
int
JyxMMix (7)
dd,
1
1
1
1
int
JyxMMiy (8)
where Mintxi and Mintyi are internal moment at nodal point i in X and Y directions,
respectively; is tensile or compressive stress parallel to grain; J is Jacobian matrix; and
are the distance between certain Gauss point and neutral axis position in and
directions, respectively. Gaussian quadrature method is employed to simplify integral
calculation of the internal stress. To adopt Gaussian quadrature method, Cartesian
coordinates (X,Y) is changed into natural coordinate (,), as illustrated in Figure 4(b).
The neutral axis position is obtained when the equilibrium condition for axial force is
attained, as written in Equation (9) and Equation (10).
iFTOTAXF (9)
dd,
1
1
1
1
JyxFTOTi (10)
where FTOTi is internal axial force at nodal point i. An equilibrium deflected shape is
obtained if the requirements in Equation (11) and Equation (12) are satisfied.
ixix MM int (11)
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.45
iyiy MM int (12)
Newton-Raphson Method for Obtaining Neutral Axis Position
Newton-Raphson method that uses the first term of Taylor is written in Equation (13). For
iterating neutral axis position, Equation (13) can be rewritten as Equation (14). FTOT’ can
be computed by using Equation (15). Iteration is started by approximating neutral axis
position ZNj. When equilibrium condition for axial force has not been obtained, ZNj is
corrected by using Equation (16). The procedure is repeated until the the internal forces is
similar to the external forces.
iiiii xxxxx 'fff (13)
CZNZNFTOTZNFTOTAXF jj
' (14)
ZN
ZNFTOTZNZNFTOTZNFTOT
jj
j
' (15)
CZNZNZN jj 1 (16)
where ZNj and ZNj+1 are neutral axis position measured from centroid of the cross section
for j and j+1 iteration, respectively; ZN is the increment of neutral axis ZN; CZN is the
correction of neutral axis position.
Observation Method
Computational Procedure
The application of Newton-Raphson method for obtaining neutral axis position is
illustrated in Figure 6.(a), where is maximum allowable relative error and MAXITZN is
the maximum number of neutral axis position iteration. Figure 6.(a) also depicts the break-
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.46
Figure 6. (a) Procedure for obtaining neutral axis position; (b) modification of neutral axis
iteration procedure
Figure 7. Applied load and nodal point
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.47
Table 1. Iteration Result for Obtaining Equilibrium Deflected Shape
Loading Lateral Load Axial Load Iteration Result Step (N) (N)
1 1000 -16000 Equilibrium deflected shape were obtained
2 2000 -16000 Equilibrium deflected shape were obtained
3 3000 -16000 Equilibrium deflected shape were obtained
4 4000 -16000 Equilibrium deflected shape were obtained
5 5000 -16000 Equilibrium deflected shape were obtained
6 6000 -16000 Equilibrium deflected shape were obtained
7 6200 -16000 Equilibrium deflected shape were not obtained
8 6500 -16000 Equilibrium deflected shape were not obtained
9 6700 -16000 Equilibrium deflected shape were not obtained
down of Figure 3 related to the calculation step of neutral axis position.
The initial value of neutral axis position ZNj and the neutral axis increment ZN are
determined using Equation (17) and Equation (18).
HZN j 05.0 (17)
20
HZN (18)
FTOT(ZNj) and FTOT(ZNj+ZN) are calculated by using Equation (10). The correction of
neutral axis position CZN is determined by using Equation (14). Maximum allowable
relative error used in iterative procedure is 0.05%. After the material property, beam
dimension, loading configuration, number of nodal point and some coefficients are defined,
program is run. Then, the identification of proper initial value for determining neutral axis
position was carried out by observing the stress distribution on cross sections that are
resulted from numerical program calculation.
Beam Dimension, Load Configuration and Other Data for Calculation
Beam dimension, load configuration, material properties and some coefficients are
required for obtaining equilibrium deflected shape and neutral axis position. Beam
dimension was 70 mm in width, 100 mm in height and 2680 mm in length. Loading
configuration and loading step are illustrated in Figure 7 and Table 1, respectively. Lateral
loads P1 were increased gradually for each calculation and the axial load was constant. The
ultimate tension and compression stress parallel to grain are 194.29 MPa and 51.22 MPa,
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.48
Table 2. Number Iteration for Obtaining Equilibrium Deflected Shape and Neutral
Axis Position
Loading Step DFL ITER at M=5
1 7 3
2 7 3
3 8 2
4 8 3
5 9 4
6 10 5
7 7 ITER > MAXITZN
8 5 ITER > MAXITZN
9 4 ITER > MAXITZN
respectively. The elasticity modulus of tension parallel to grain is 17,200 MPa. The non
linearity of compressive stress-strain curve is represented by Equation (19).
150.02204810.2 //
2
//
6
// ccc (19)
where σc// and εc// are the compressive stress and strain parallel to grain, respectively.
Number of nodal point M defined was 9 (see Figure 5). Number of gauss point in X
direction was 5. Number of gauss point in Y direction was 5. Maximum number of
equilibrium deflected shape iteration MAXITDFL is 30. Maximum number of neutral axis
position iteration MAXITZN is 25.
Result and Discussion
The Iteration Result of Equilibrium Deflected Shape and Neutral Axis Position
In general, the iteration result of equilibrium deflected shape can be seen in Table 1.
Iteration number needed to get equilibrium deflected shape DFL can be seen in Table 2.
Further, Table 2 also depicts iteration number needed to get neutral axis position ITER at
nodal point 5 for each loading step.
Referring to Table 1, the equilibrium deflected shape can be obtained for loading step 1
up to 6. It also means that the iteration for defining neutral axis position get convergent
result for loading step 1 up to 6. The neutral axis position is achieved when the equilibrium
condition for axial force is attained. Then, when internal moments are equal to external
moments, it will lead to achievement of equilibrium deflected shape.
Moreover, Table 1 and Table 2 show that the equilibrium deflected shape cannot be
obtained when the beam was subjected to loading step of 7, 8, and 9. It is caused by the
iteration number for obtaining neutral axis position ITER exceeds the allowable maximum
number of neutral axis position iteration MAXITZN. Further, the question is why the
neutral axis position cannot be resulted. It is needed to know whether the iteration has
correctly stopped or not. Thus, the observation was carried out on a cross section at nodal
point 5 (see Figure 7) which has the maximum tensile and compressive stress.
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.49
Figure 8. Stress distribution on a cross section located at nodal point M = 5 for load
number: (a) 1; (b) 2; (c) 3; (d) 4; (e) 5; (f) 6
Figure 9. (a) Gauss point position on a cross section; (b) stress distribution at nodal point 5,
DFL = 7, ITER = 1
The Result of Stress Distribution Observation
Figure 8 depicts the stress occurred at Gauss point, uppermost side, as well as bottommost
side of a cross section that is located at nodal point 5. The location of Gauss point on a
cross section can be seen in Figure 9.(a).
As illustrated in Figure 8, the neutral axis position at nodal point 5 is 10.115 mm, 4.244
mm, 2.592 mm, 1.946 mm, 1.830 mm and 2.307 mm when the structure was accordingly
subjected to loading step 1 to 6. It depicts that the neutral axis position tends to move
upward when the lateral load is increased while the axial load is constant. In case of
loading step 6, maximum tension and compression parallel to grain are 53.913 MPa and
49.817 MPa, respectively. According to the mechanical properties of glulam bamboo, the
ultimate tension and compression stress are 194.29 MPa and 51.22 MPa, respectively. It
means that the structure had not failed yet, neither in tension or compression.
When the load was increased, as can be seen in Table 2, equilibrium deflected shape
cannot be obtained because iteration number of neutral axis position ITER is greater than
the allowable maximum number of iteration for neutral axis position MAXITZN. This
reason causes the iteration of neutral axis position cannot achieve convergent result.
Regarding structure subjected to axial compression and lateral load as well as the
assumption that the stress is zero when it is greater than the ultimate stress, non convergent
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.50
Table 3. Neutral Axis Position ZN at Nodal Point M = 5 for Every DFL When
Structure is Loaded with Load Number of 7
DFL ZN (mm) DFL ZN (mm) DFL ZN (mm)
1 59.272 4 2.004 7 Stop at M=5
2 1.817 5 2.067 - -
3 1.915 6 2.105 - -
Table 4. Neutral Axis Position ZN for Every Iteration of Neutral Axis Position ITER
When Structure is Loaded with Load Number of 7 at Nodal Point M = 5, Iteration
Number of Equilibrium Deflected Shape DFL = 12
ITER ZN (mm) ITER ZN (mm) ITER ZN (mm)
1 5 11 7.258 21 17.291
2 20.128 12 28.488 22 7.256
3 6.523 13 20.47 23 28.479
4 25.738 14 6.422 24 20.471
5 35.323 15 25.407 25 6.422
6 18.05 16 34.851 26 25.406
7 7.128 17 18.127
8 28.002 18 7.115
9 38.448 19 27.95
10 17.276 20 38.376
result will take place if compressive stress is greater than ultimate stress of material.
Table 2 shows that iterative calculation for obtaining equilibrium deflected shape of
structure that was subjected to loading step 7 stopped when the iteration number of
equilibrium deflected shape DFL is equal to 7. In detail, the iteration can be seen in Table
3. It informs that, the neutral axis position tends to be located around 2 mm from the
centroid of cross section. Referring to Figure 8, this location comes close to the neutral
axis position for the previous loading when equilibrium deflected shape was obtained.
However, in case of loading step 7, the iteration stopped before the neutral axis position
was determined. Accordingly, further observation was needed to figure out whether the
internal compression stress is greater than the ultimate compression stress of the material.
The observation was conducted by investigating neutral axis position resulted from
every iteration ITER, as shown in Table 4. It is shown that the uppermost position of
neutral axis is 5 mm, resulted from iteration number one. This stress distribution can be
seen in Figure 9.(b). It depicts that the maximum compressive and tensile stress are 51.203
MPa and 52.473 MPa, respectively. It implies that the both stress are less than the ultimate
stress of material. Referring to Figure 8 and Table 3, if the equilibrium deflected shape of
the structure subjected to loading step 7 can be obtained, the neutral axis will be located
around 2 mm from the centroid of a cross section. Concerning Figure 9.(b), the
compressive stress in which the neutral axis is located around 2 mm is less than that in
which the neutral axis is located at 5 mm. It can be concluded that the ultimate
compression stress of the material has not been achieved when the structure is subjected to
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.51
Table 5. Iteration Result for Obtaining Equilibrium Deflected Shape After
Modification on Neutral Axis Iteration Procedure
Loading Lateral Load Axial Load Iteration Result Step (N) (N)
1 1000 -16000 Equilibrium deflected shape were obtained
2 2000 -16000 Equilibrium deflected shape were obtained
3 3000 -16000 Equilibrium deflected shape were obtained
4 4000 -16000 Equilibrium deflected shape were obtained
5 5000 -16000 Equilibrium deflected shape were obtained
6 6000 -16000 Equilibrium deflected shape were obtained
7 6200 -16000 Equilibrium deflected shape were obtained
8 6500 -16000 Equilibrium deflected shape were obtained
9 6700 -16000 Equilibrium deflected shape were not obtained
loading step 7. In line with this conclusion, the iteration for neutral axis position should
achieve convergent result. The non convergent result might be caused by inappropriate
initial value for neutral axis position.
Modifying Initial Value for Neutral Axis Iteration
According to previous observation result, initial approximation of neutral axis position
presented in Equation (17) is modified to Equation (20).
1
1 ,1 to2for
,05.0
DFL
DFL
MiiNAP
HZN j (20)
where NAP is array of neutral axis position that is defined from previous iteration for each
nodal point.
In principle, the iteration results of neutral axis position are stored in variable array
NAP. Then, if the equilibrium deflected shape has not been achieved, the calculation is
continued by applying previous neutral axis position that is stored in variable array NAP as
the next initial value. Its application on numerical program can be seen in Figure 6.(b).
Besides the improvement of initial approximation, the increment of neutral axis ZN is
modified as follows.
1
1
,50
,20
DFL
DFL
H
HZN (21)
The modification aims to increase the accuracy for each step of neutral axis iteration.
Using Modified Initial Neutral Axis Position and Its Increment
Table 5 demonstrates the calculation results after initial value of neutral axis position and
its increment have been modified. It shows that equilibrium deflected shape can be
obtained for loading step 1 up to 8 when the initial value of neutral axis position is defined
as Equation (20). As previously described, it also means that neutral axis position can be
obtained when the initial neutral axis position written in Equation (20) is used. Further,
besides getting convergent result on the iteration of equilibrium deflected shape and neutral
axis position, the number of iteration ITER needed to achieve neutral axis position after
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.52
Figure 10. Stress distribution on a cross section located at nodal point M = 5 after
modification of initial neutral axis position for load number:
(a) 1; (b) 2; (c) 3; (d) 4; (e) 5; (f) 6; (g) 7; (h) 8
Table 6. Number Iteration for Obtaining Equilibrium Deflected Shape and Neutral
Axis Position After Modification on Neutral Axis Iteration Procedure
Loading DFL
ITER Load DFL
ITER
Step at M=5 Number at M=5
1 7 2 6 10 2
2 7 2 7 10 2
3 8 2 8 11 2
4 8 2 9 25 ITER > MAXITZN
5 9 2
modification is less than that before modification, as depicted in Table 6. It indicates that
the modification also can speed up calculation.
The stress distribution and neutral axis position on a cross section located at nodal
point 5 can be seen in Figure 10. In case of loading step 7, the compressive and tensile
stresses are 50.518 MPa and 56.137 MPa, respectively. It proves the previous conclusion
stating that the maximum compressive strength of the material has not been achieved
Figure 10 illustrates that when structure is loaded with loading step 8, the stress on
uppermost side has exceeded the ultimate stress of material. As load is increased to load
number of 9, internal stress is greater. It can cause the equilibrium condition for axial force
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.53
cannot be obtained. Hence, the equilibrium deflected shape cannot be resulted, as depicted
in Table 5. In line with this reason, the iterative program is already correct. It means that
Equation (20) can be considered as appropriate initial approximation for Newton-Raphson
iteration to define neutral axis position.
Conclusion
In this study, the appropriate initial approximation for Newton-Raphson method to obtain
the neutral axis position of glulam bamboo has been observed. The assumption used for
calculating internal forces is that the tension and compression parallel to grain are equal to
zero when they are greater than its ultimate stress. Glulam bamboo is assumed as
homogeneous material. Based on the observation, Equation (20) can be considered as an
appropriate initial value for neutral axis position while Equation (21) is applied for
calculating neutral axis increment.
Acknowledgments
The authors would like to acknowledge to Prof. Morisco and Dr. Fitri Mardjono
(Department of Civil and Environmental Engineering, Gadjah Mada University, Indonesia)
for great supervision during the research. The authors also would like to thank to Prof.
Akinori Tani and Prof. Yuichiro Yamabe (Department of Architecture Engineering, Kobe
University, Japan) for the facility given during accomplishing this article.
References
[1] S.M. Rittironk, and Elnieri, “Investigating laminated bamboo lumber as an alternate to
wood lumber in residential construction in The United States.” Modern Bamboo
Structures, Taylor and Francis Group, London, UK, pp. 83-96, 2008.
[2] Morisco, Rekayasa Bambu, Nafiri, Yogyakarta, Indonesia, 1999.
[3] P. Kornerup, and J.M. Muller, “Choosing starting values for certain Newton-Raphson
iteration.” Theoritical Computer Science, 351, pp. 101-110, 2006.
[4] P.V.R. Murthy, and K.P. Rao, “Finite element analysis of laminated anisotropic beams
of bimodulus materials.” Computers and Structures, 18-5, pp. 779-787, 1984.
[5] M. Brunner, and M. Schnueriger, “Timber beams strengthened by attaching prestressed
carbon FRP laminates with a gradiented anchoring device.” Proc. International
Symposium on Bond Behaviour of FRP in Structures, International Institute for FRP in
Construction, pp. 465-471, 2005.
[6] R. Mu, J.P. Forth, A.W. Beeby, and R. Scott, “Modelling of shrinkage induced
curvature of cracked concrete beams.” Tailor Made Concrete Structure, Taylor and
Francis Group, London, UK, pp. 573-578, 2008.
[7] T.E. Shoup, Numerical methods for the personal computer, Prentice-Hall, New Jersey,
US, 1983.
[8] W.J. Gilbert, “Generalization of Newton’s Method.” Fractals, World Scientific
Publishing Company, 9-3, pp. 251-262, 2001.
[9] W. Chen, and T. Atsuta, Theory of Beam-Columns In-Plane Behaviour and Design,
McGraw-Hill, United States of America, 1976.
ASEAN Engineering Journal Part C, Vol 3 No 1, ISSN 2286-8150 p.54